Tomáš Neustupa | Searcharxiv

arXiv2020

The weak Stokes problem associated with a flow through a profile cascade in Lr-framework

We study the weak steady Stokes problem, associated with a flow of a Newtonian incompressible fluid through a spatially periodic profile cascade, in the Lr-framework. The used mathematical model is based on the reduction to one spatial period, represented by a bounded 2D domain Omega. The corresponding Stokes problem is formulated by means of three types of boundary conditions: the conditions of periodicity on the "lower" and "upper" parts of the boundary, the Dirichlet boundary conditions on the "inflow" and on the profile and an artificial "do nothing"--type boundary condition on the "outflow". Under appropriate assumptions on the given data, we prove the existence and uniqueness of a weak solution in W^{1,r}(Omega) and its continuous dependence on the data. We explain the sense in which the "do nothing" boundary condition on the "outflow" is satisfied.

math.AP↗

arXiv2020

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in $L^r$-framework

The paper deals with the Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. We use results from [32] (the maximum regularity property in the $L^2$-framework) and [33] (the weak solvability in $W^{1,r}$), and extend the findings on the maximum regularity property to the general $L^r$-framework (for $1<r<\infty$). Using the reduction to one spatial period $Ω$, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $Γ_0$ and $Γ_1$, the Dirichlet boundary conditions on $Γ_{\rm in}$ and $Γ_P$ and an artificial "do nothing"-type boundary condition on $Γ_{\rm out}$ (see Fig. 1). We show that, although domain $Ω$ is not smooth and different types of boundary conditions "meet" in the vertices of $\partialΩ$, the considered problem has a strong solution with the maximum regularity property for "smooth" data. We explain the sense in which the "do nothing" boundary condition is satisfied for both weak and strong solutions.

math.AP↗